Using hierarchical spatial data structures for hierarchical spatial reasoning

This paper gives a definition of Hierarchical Spatial Reasoning; which computes increasingly better results in a hierarchical fashion and stops the computation when a result is achieved which is ‘good enough’. This is different from standard hierarchical algorithms, which use hierarchical data structures to improve efficiency in computing the correct result. An algorithm on hierarchical spatial data structure explores all details where such exist. An hierarchical reasoning algorithm stops processing if additional detail does not effectively contribute to the result and is thus more efficient. Hierarchical spatial data structures, especially quadtrees, are used in many implementations of GIS and have proved their efficiency. Operations on hierarchical spatial data structures are effective to compute spatial relations. They can be used for hierarchical spatial reasoning. A formal definition of hierarchical spatial reasoning requires• a coarsening function c, which produces a series of less detailed representations from a most detailed data set,• a function of interest f which is applicable to these representations, and• a function f which computes for each representation the quality of the result. The computation starts with the least detailed representation and continues till a result with sufficient quality is found. This is demonstrated with a simplified example, based on a raster computation for area computation and overlap. Examples from the literature demonstrate that the same hierarchical aggregation and similar coarsening functions can be used for a wide variety of spatial reasoning tasks.